Derivability structures
type: Item
instance of: scholarly article
title: Derivability structures
zbMATH Open document ID: 1149.18009
author: Georges Maltsiniotis
author: Bruno Kahn
published in: Advances in Mathematics
publication date: 2008-06-09T00:00:00Z
review text: The authors introduce and compare different types of ``derivability structures'' for categories \(\mathcal C\) equipped with a class of morphisms \(W\) (which are thought of as weak equivalences). Such structures are given by a category \(\mathcal C_0\) together with weak equivalences \(W_0\) and a functor \(K: \mathcal C_0 \rightarrow \mathcal C\) sending \(W_0\) to \(W\). One often assumes that \(W_0\) contains the identities and is stable under composition. The idea is that \(K\) will allow for the construction of certain resolutions which then permit to establish existence theorems for derived functors. For example in a simplicial derivability structure one requires that the category of \(K\)-resolutions of any object \(c \in \mathcal C\), i.e. the category of weak equivalences from \(c\) to objects of the form \(K(d)\), be \(1\)-connected, that of \(K\)-resolutions of any morphism be connected, and that of \(K\)-resolutions of any pair of composable morphisms be non-empty. The main results are that the derivability structures introduced by \textit{W. G. Dwyer, P. S. Hirschhorn, D. M. Kan}, and \textit{J. H. Smith} [``Homotopy limit functors on model categories and homotopical categories'', Math. Surv. Monogr. 113 (2004; Zbl 1072.18012)], as well as the ``Anderson-Brown-Cisinski'' structures are simplicial, whereas the structures appearing in the work of Radulescu-Banu are derivability structures but probably not of simplicial type.
reviewed by: Jérôme Scherer
Mathematics Subject Classification ID: 18G55
Mathematics Subject Classification ID: 55U35
Mathematics Subject Classification ID: 18A40
Mathematics Subject Classification ID: 18E35
Mathematics Subject Classification ID: 18G10
Mathematics Subject Classification ID: 55P60
zbMATH DE Number: 5285714
zbMATH Keywords: homotopical algebra
zbMATH Keywords: derived functor
zbMATH Keywords: model category
MaRDI profile type: Publication
full work available at URL: j.aim.2008.03.010
OpenAlex ID: W1997922305
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DOI: 10.1016/J.AIM.2008.03.010
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description: scientific article; zbMATH DE number 5285714